Several elementary functions which are defined via power series may be defined in any unital Banach algebra ; examples include the exponential function and the trigonometric functions, and more generally any entire function.

In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial.

* History of elementary algebra

As an introduction, elementary algebra can be found in books from the early 19th century.

The mathematical definition of an elementary function, or a function in elementary form, is considered in the context of differential algebra.

The harmonic mean also features in elementary algebra when considering problems of working in parallel.

Linear equations can be rewritten using the laws of elementary algebra into several different forms.

This is elementary linear algebra – one direction is trivial ; the other follows from:

If the subsets of X in Σ correspond to numbers in elementary algebra, then the two set operations union ( symbol ∪) and intersection (∩) correspond to addition and multiplication.

Though coefficients are frequently viewed as constants in elementary algebra, they can be variables more generally.

The idea of the hyperreal system is to extend the real numbers R to form a system * R that includes infinitesimal and infinite numbers, but without changing any of the elementary axioms of algebra.

The exponentiation operation with integer exponents requires only elementary algebra.

In abstract algebra and logic, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.

While Escalante teaches basic arithmetic and elementary and intermediate algebra, he realizes that his students have far more potential.

This method generalizes to systems with additional variables ( see " elimination of variables " below, or the article on elementary algebra.

Peacock's principle may be stated thus: the elementary symbol of arithmetical algebra denotes a digital, i. e., an integer number ; and every combination of elementary symbols must reduce to a digital number, otherwise it is impossible or foreign to the science.

If the former horn of the dilemma is chosen, arithmetical algebra becomes a mere shadow ; if the latter horn is chosen, the operations of algebra cannot be defined on the supposition that the elementary symbol is an integer number.

It can be proven with elementary algebra and holds in every commutative ring.

Since 1968 there is the Risch algorithm for determining indefinite integrals that can be expressed in term of elementary functions, typically using a computer algebra system.

In elementary algebra, when solving equations, it is " guess and check ".

Similarly, Fermat's last theorem is stated in term of elementary arithmetic, which is a part of commutative algebra, but its proof involves deep results of both algebraic number theory and algebraic geometry.

A typical application is furnished by the Arzelà – Ascoli theorem and in particular the Peano existence theorem, in which one is able to conclude the existence of a function with some required properties as a limiting case of some more elementary construction.

An elementary student wearing the typical white and red uniform in Indonesia.

Voiced by Trey Parker, Cartman is an obese, immature, spoiled, selfish, manipulative, lazy, foul-mouthed, deluded, mean-spirited, sadistic, racist, sexist, anti-semitic, homophobic, xenophobic, sociopathic, narcissistic, and ill-tempered elementary school student living with his unwed mother in the fictional town of South Park, Colorado, where he routinely has extraordinary experiences not typical of conventional small-town life.

But because of increasing specialization and depth in the student's major field of study, a typical core curriculum in higher education mandates a far smaller proportion of a student's course work than a high school or elementary school core curriculum prescribes.

Another typical aspect of the Aguarunas consists in the fact that they have traditionally worked as a seminomadic population, due to the poverty of the agricultural soil and the extremely elementary, traditional agrarian technology, which brings as a consequence the depletion of the soil in a short period of two or three years.

As a musical term, sonatina has no single strict definition ; it is rather a title applied by the composer to a piece that is in basic sonata form, but is shorter, lighter in character, or more elementary technically than a typical sonata.

In mathematics, the Heilbronn triangle problem is a typical question in the area of irregularities of distribution, within elementary geometry.

Carl Friedrich von Weizsaecker identified 10 < sup > 120 </ sup > with the ratio of the universe's volume to the volume of a typical nucleon bounded by its Compton wavelength, and he identified this ratio with the sum of elementary events or bits of information in the universe.

Cardinal Carter is a unique school with unique needs and concerns, not a typical elementary or secondary school but a hybrid of both, serving students from grade 7 to 12.

Primary education in PISD, following the typical U. S. structure, consists of 44 elementary schools that serve the kindergarten through fifth grades.

The following are examples of elementary events:

Some elementary examples of groups in mathematics are given on Group ( mathematics ).

( Item 6 under Further examples expresses the same U in less elementary language via presheaves.

* A. C. Dixon The elementary properties of the elliptic functions, with examples ( Macmillan, 1894 )

All examples above are elementary amenable.

To give them something relevant to study, he began using examples from the Apollo program as applications of elementary physics.

# Transform familiar information examples into elementary facts, and apply quality checks

Some further elementary examples can be constructed from these by using the obvious fact that the product K ( G, n ) × K ( H, n ) is K ( G × H, n ).

The software is provided with about 2, 000 tests: they are devoted to elementary qualification and are useful as examples.

found examples of graphs for which M does indeed grow very fast and is at least as large as a 1 / ε < sup > 1 / 16 </ sup >- level iterated exponential of m. In particular the best bound has level exactly 4 in the Grzegorczyk hierarchy, and so is not an elementary recursive function.